J. Phase Equilib. Diffus. https://doi.org/10.1007/s11669-019-00736-w

Multiscale and Its Implications to Critical Phenomena, Emergent Behaviors, and Information

1 2 3 Zi-Kui Liu • Bing Li • Henry Lin

Submitted: 16 January 2019 / in revised form: 28 May 2019 Ó ASM International 2019

Abstract Thermodynamics of critical phenomena in a stable configuration. Furthermore, the connection between system is well understood in terms of the divergence of change of configurations and the change of information is molar quantities with respect to potentials. However, the discussed, which provides a quantitative framework to prediction and the microscopic mechanisms of critical study complex, dissipative systems. points and the associated property anomaly remain elusive. It is shown that while the critical point is typically con- Keywords critical phenomena Á entropy Á information Á sidered to represent the limit of stability of a system when invar Á perovskites Á second law of thermodynamics Á the system is approached from a homogenous state to the statistic thermodynamics critical point, it can also be considered to represent the convergence of several homogeneous subsystems to become a macro-homogeneous system when the critical point is approached from a macro-heterogeneous system. 1 Introduction Through the understanding of statistic characteristics of entropy in different scales, it is demonstrated that the Thermodynamics is a science concerning the state of a statistic competition of key representative configurations system described by a set of state variables. Entropy is one results in the divergence of molar quantities when of the state variables, representing the degree of disorder of metastable configurations have higher entropy than the the system, i.e., the higher the disorder, the larger the entropy.[1] While the first law of thermodynamics is on the energy conservation, the second law of thermodynamics This article is an invited paper selected from presentations at ‘‘PSDK dictates that any internal process (IP or ip) in a system must XIII: Phase Stability and Diffusion Kinetics,’’ held during MS&T’18, produce entropy if it proceeds spontaneously and irre- October 14-18, 2018, in Columbus, Ohio. The special sessions were dedicated to honor Dr. John Morral, recipient of the ASM versibly. It should be noted though that the total entropy International 2018 J. Willard Gibbs Phase Equilibria Award ‘‘for change of the system also depends on how the system fundamental and applied research on topology of phase diagrams and exchanges entropy with the surroundings through heat and theory of phase equilibria resulting in major advances in the mass and can be either positive or negative, and it is the calculation and interpretation of phase equilibria and diffusion.’’ It has been expanded from the original presentation. combined first and second laws of thermodynamics that represents the over-all progress of the system. The degree & Zi-Kui Liu of order or disorder of a system may thus either increase or [email protected] decrease based on the external conditions and internal 1 Department of Materials Science and Engineering, The processes, resulting in the change of the state of the system Pennsylvania State University, University Park, PA 16802 in terms of internal configurations and their probabilities 2 Department of Statistics, The Pennsylvania State University, that are denoted by the configurational entropy at the cor- University Park, PA 16802 responding time and space scales. 3 Department of Ecosystem Science and Management, The Each of those configurations can be considered as a sub- Pennsylvania State University, University Park, PA 16802 system itself with its own set of sub-configurations. For 123 J. Phase Equilib. Diffus. example, one can investigate the entropy of the universe where T, P and V are temperature, pressure, and volume, and a black hole,[2,3] a society,[4,5] an ecosystem,[6] a per- [7] [8] respectively, and li is the chemical potential of component son or a compound, and the entropy from the smaller i. In the second part of Eq 2, Ya denotes the potentials, i.e., scale is homogenized and contributes to the entropy at the a T, - P and li, and X denotes the molar quantities, i.e., S, larger scale. The present paper aims to discuss how the V and N .[11,12] It should be emphasized that both dV and homogenization of configurations can be formulated and i dNi in Eq 2 refer to the changes between the system and exchanged between scales in terms of configurational surroundings, while dS contains the contributions from IPs at different scales, its probability at neighboring as shown by Eq 1. scale and its application to systems with critical points. By introducing the driving force for each independent Additionally, the entropy production of an internal process IP, j, the energy change due to the entropy production can is correlated with the generation and erasure of information be represented by the product of driving force for the IP, and reflects the information stored in the system in terms of Dip;j, and the change of corresponding internal variable, multiscale configurations. It is noted that the present work dn , in terms of the Taylor expansion up to the third order is closely related to the theories[9,10] with j as follows[11] the system at one scale consisting of self-similar copies of X X itself when viewed at another scale, and the different 1 TdipS ¼ Dip;jdnj À Dip;jkdnjdnk parameters at various scales are used to describe the con- 2 1 X stituents of the system. It is shown in the present work that þ D dn dn dn ðEq 3Þ 6 ip;jkl j k l the entropy and statistical probability of each configuration are the parameters that connect the scales. where the second and third terms are added for discussion of stability of the system based on their signs when the first summation in the equation equals zero, i.e. the system is at 2 Review of Fundamentals of Entropy a state of equilibrium.[13] The second law of thermodynamics requires that each In thermodynamics, the entropy change of a system, dS, independent IP, if proceeding spontaneously, must have a [11,12] can be written as follows positive entropy production, i.e. Dip;j [ 0 to give dQ X TdipS [ 0. Consequently, the system reaches a state of dS ¼ þ SidNi þ dipS ðEq 1Þ D T equilibrium when ip;j  0 for all IPs. This equilibrium state is stable with respect to fluctuations of internal vari- where dQ and dNi are the heat and the amount of com- ables when Dip;jk [ 0 due to the negative entropy produc- ponent i that the system receives from or release to the tion and unstable when Dip;jk\0 due to the positive entropy surroundings, T is the temperature, Si is the molar entropy production for the IP of interest, both as shown by Eq 3. of component i in the surroundings for dNi [ 0 or the When Dip;jk ¼ 0 the system is at the limit of stability. The system for dN \ 0, often called partial entropy of com- i limit of stability becomes a critical point in the space of ponent i, and d S is the entropy production due to inde- ip independent internal variables n n n with additional pendent IPs with each that may contain a group of coupled j k l D ¼ 0. The critical point in the space of all possible processes. It is evident that the first two terms concern the ip;jkl independent internal variables of the system is termed as interactions between the surroundings and the system, the invariant critical point (ICP).[14] For a homogeneous while the third term embodies what happens inside the system, the IPs involve the movement of molar quantities system. Equation 1 thus establishes a bridge connecting the inside the system, and one can write the stability variables interior of a system in terms of internal entropy production of one IP as follows[11] and the exterior of the system in terms mass and heat "# exchanges. o2U oYa Dip;XaXa ¼ ¼ ðEq 4Þ Combining Eq 1 with the first law of thermodynamics, a 2 o a o X X b ðÞXb X the combined law of thermodynamics can be obtained. "#"# While the work exchanges between the system and sur- o3U o2Ya Dip;XaXaXa ¼ ¼ ðEq 5Þ rounding can involve mechanical, electric, and magnetic oðÞXa 3 oðÞXa 2 works, it is often that the work due to hydrostatic pressure Xb Xb is considered,[11,12] and the combined law of thermody- For heterogeneous systems with chemical reactions, the namics is written as follows IPs are more complicated and may or may not be explicitly X spelled out.[15] dU ¼ TdSX À PdV þ lidNi À TdipS a a Equation 1 concerns the change of entropy. To obtain ¼ Y dX À TdipS ðEq 2Þ the absolute value of entropy, one can integrate the

123 J. Phase Equilib. Diffus. equation under the condition of reversible addition of heat to the system in equilibrium with dNi ¼ 0 and dipS ¼ 0, i.e. CdT S ¼ S þ r T ðEq 6Þ 0 0 T where S0 is the entropy at zero Kelvin, conventionally assigned to be zero in terms of the third law of thermo- dynamics, and C the heat capacity of the system, denoting the heat needed to increase the temperature of the system by one degree.

Fig. 1 Two scenarios of configurations of a system, (a) k 2 fg1; ...; m configurations with each of them composed of l 2 3 Statistics of Entropy fg1; ...; n sub-configurations; (b) l 2 fg1; ...; n configurations with each being the statistic average of each l 2 fg1; ...; n sub-configu- The discussion in the previous section has not concerned ration in all k 2 fg1; ...; m configurations the statistic characteristics of entropy. Gibbs[16] pointed out that the entropy is defined as the average value of the logarithm of probability of phase where the differences in This equation can be extended in the directions of larger phases are with respect to configuration. Therefore, the and smaller dimensions to capture more complexity of the configurational entropy in a system of interest with prop- system or in the directions of longer and shorter time to erly defined time and space scales can be written as capture evolution of the system. The examples of config- urations k 2 fg1; ...; m and l 2 fg1; ...; n are the magnetic Xm conf k k spin and vibrational configurations for cerium and Fe3Pt, S ¼ÀkB p ln p ðEq 7Þ k¼1 and the atomic and vibrational configurations for the mis- cibility gap in the fcc Al-Zn solution, respectively, dis- k where kB is the Boltzmann constant, and p the probability cussed in section 6. ofP configuration k 2 fg1; ...; m of the system with On the other hand, from a statistics point of view, one m k k¼1 p ¼ 1 as shown in the upper row of Fig. 1(a). Since can calculate the direct contributions from the configura- each configuration k has its own entropy, Sk, the total tions in the scale l to the system as follows by re-organizing entropy of the system can be written as Eq 10 with the joint probability defined as pk;l ¼ pkpljk, Xm Xm ÀÁ Xm Xn  k k conf k k k k ljk ljk k ljk S ¼ p S þ S ¼ p S À kB ln p ðEq 8Þ S ¼ p p S À kB ln p p k¼1 k¼1 k¼1 l¼1 Xm Xn  k;l ljk k;l Equations 6 and 8 should give the same entropy value ¼ p S À kB ln p ðEq 11Þ when they are counted on the same time and space scales. k¼1 l¼1 By the same token, the configuration k is composed of It is self-evident that Eq 11 is the same as the combi- configurations in the scale of short time and smaller P nation of Eq 8 and 9 due to n pljk ¼ 1 and dimension and can be written as P ÀÁ l¼1 n pljkk ln pk ¼ k ln pk. Furthermore, one may Xn  l¼1 B B k ljk ljk ljk attempt to switch the order of summation of k and l, i.e., S ¼ p S À kB ln p ðEq 9Þ "# ! l¼1 Xn Xm Xm S ¼ pk;lSljk À k pk;l ln pk;l ðEq 12Þ where pljk and Sljk are the conditional probability and B l¼1 k¼1 k¼1 entropy of configuration l 2 fg1; ...; n as sub-configura- Pn This is analogue to Eq 10, but with the configuration ljk tions of configuration k with p ¼ 1 as shown in the l 2 fg1; ...; n with sub-configuration of k 2 fg1; ...; m as l¼1 lower row of Fig. 1(a). Equation 8 can then be re-orga- shown in Fig. 1(b). As proved below, the two scenarios in nized as follows Fig. 1(a) and (b) indeed give the same entropy of the !system. Xm Xn  k ljk ljk ljk k Let us define the following S ¼ p p S À kB ln p À kB ln p Xm k¼1 l¼1 ql ¼ pkpljk ðEq 13Þ Eq 10 ð Þ k¼1

123 J. Phase Equilib. Diffus.

pkpljk pkpljk pk;l statistic averaging is carried out. The implications of this qkjl P Eq 14 ¼ m k ljk ¼ l ¼ l ð Þ conclusion will be further discussed in section 6. k¼1 p p q q Xm  l kjl ljk kjl T ¼ q S À kB ln q ðEq 15Þ k¼1 4 Probability of Configurations P P n l m kjl l where l¼1 q ¼ 1, k¼1 q ¼ 1, T is the entropy of a Probability of individual configurations of a system at configuration in l 2 fg1; ...; n with sub-configuration of given time and space scales can be evaluated when their k 2 fg1; ...; m (see Fig. 1b), and Eq 14 the commonly energetics are known at the corresponding time and space [17] l referred Bayes’s theorem. It can be seen that T consists scales through their partition functions. For a closed system of two parts: a) the entropyP of each configuration in the under constant temperature and volume, a collection of m kjl ljk lower row of Fig. 1(b), i.e. k¼1 q S which is evaluated possible configurations of the system comprises a canonical from the entropy of each configuration in l 2 fg1; ...; n for ensemble. In the discrete form, the canonical partition [18] all configurations in k 2 fg1P; ...; m , and b) the configura- function is defined as follows m kjl kjl tion among them, i.e. Àk q ln q . X X k B k¼1 k À F The proof for the fact that the two scenarios schemati- Z ¼ Z ¼ e kBT ðEq 19Þ cally depicted in Fig. 1(a) and (b) have the same entropy is k k as follows. where Zk and Fk are the partition function and Helmholtz energy of configuration k with Theorem If k k Xm ÀÁ F ¼ÀkBT ln Z ðEq 20Þ k k k S ¼ p S À kB ln p ðEq 16Þ k¼1 It should be mentioned that in the literature, the internal or total energy, Uk, is often used in the place of Fk, which then implicitly assumes that the entropy of each configuration is Xn ÀÁ negligible. This assumption is not valid at high tempera- l l l S ¼ q T À kB ln q ðEq 17Þ tures as the individual configurations can have different l¼1 entropy values resulting in significant change of their Proof From Eq 11 or 12 and using Eq 13 to 15 respective probabilities. It is noted that Asta et al.[19] used Xn Xm  an equation similar to Eq 19 for systems under constant k;l ljk k;l S ¼ p S À kB ln p temperature, pressure, and chemical potentials. l¼1 k¼1 The probability of each configuration can be defined as Xn Xm  l kjl ljk l kjl k k ¼ q q S À kB ln q q k Z FÀF p ¼ ¼ e kBT ðEq 21Þ l¼1 k¼()1 Z Xn Xm  l kjl ljk l kjl where F is the Helmholtz energy of the system and can be ¼ q q S À kB ln q q written as follows[8,20] l¼1 ()"#k¼1 Xn Xm Xm X X l kjl ljk kjl l kjl F k T ln Z pkFk k T pk ln pk Eq 22 ¼ q q S À kB ln q À kB ln q q ¼À B ¼ þ B ð Þ l¼1 k¼1 k¼1 k k Xn ÀÁ It can be seen that the configurational entropy by Eq 7 is ¼ ql Tl À k ln ql B shown in the last term in Eq 22, and from Eq 21 that l¼1 P F Fk for pk 1, originated from pk ln pk 0. The ðEq 18Þ   k  equality holds when the system has only one configuration. The above discussion does not consider interactions between configurations. However, when the fluctuation This proof is important as it demonstrates that the dimension is smaller than the dimension of the system, sequence of configuration averaging can be switched in there are interactions between configurations that may terms of their scales, i.e. either at the scale k 2 fg1; ...; m result in new configurations that are not part of the existing or l 2 fg1; ...; n with the other scale as its sub-configura- configurations. These interactions may be considered tions as schematically depicted in Fig. 1(a) and (b). Pro- explicitly by adding interaction terms in Eq 22 similar to vided individual and joint probabilities of configurations in the CALPHAD modeling method in thermodynamics in the the system can be evaluated, the entropy of the system j k jk jk remains invariant independent of the scale where the form of p p L with L being the interaction

123 J. Phase Equilib. Diffus.

[11,21,22]  energy. On the other hand, a better approach is to oS X opk ÀÁoSk k opk ¼ Sk À k ln pk þ pk À B expand the set of configurations to include the new con- o o B o k o T k T T p T figurations with the statistic approach intact as discussed in oSN this paper. ¼ oT  Furthermore, experimental measurements can only X opk pk oSk oSN þ Sk À SN À k ln þ pk À detect the combined effect from all configurations. For the oT B pN oT oT properties of individual configurations, one has to rely on k6¼N theoretic calculations. The first-principles quantum ðEq 26Þ mechanics technique based on the density functional theory [23] where N denotes the configuration with the lowest Helm- (DFT) along with the sophisticated computer pro- holtz energy, i.e., the ground state at zero Kelvin, and [24,25] P P grams and ubiquitous high performance comput- k opk oSN [26,27] p 1 and 0 are used. In Eq 26, is posi- ers have enabled the quantitative predictions of k ¼ k oT ¼ oT ground states of a vast configurations as reflected in a tive from Eq 4, and the summation in Eq 26 would also be k N k number of online databases.[28–30] The Helmholtz energy positive if S [ S that results in the increase of p from N of a configuration can be effectively evaluated from either zero at zero Kelvin at the expense of p . phonon calculations[31,32] or the Debye model[33] with the Differentiation of Eq 21 yields [34] scaling factor obtained from the elastic properties pre- opk pk ÂÃÀÁÀÁ [35–37] ¼ Fk À F þ TSk À S dicted from first-principles calculations. For config- o 2 T kBÀÁT  urations with disordering, the cluster expansion pk Sk À S Fk À F [38,39] approach or the special quasirandom structures ¼ 1 þ k ðEq 27Þ kBT TSðÞÀ S (SQS)[40–42] can be used. It is also possible to sample opk configurations at finite temperatures through the ab initio Equation 27 further demonstrates that oT [ 0 due to molecular dynamics (AIMD) calculations[43] with the Sk [ S with S  SN at the limit of pk near zero and Fk [ F. oS atomic forces computed on the fly using the DFT-based It is evident that a dramatic increase of oT has to come from o k first-principles calculations as recently demonstrated for the dramatic increase of p , i.e., the significant competition [44,45] oT PbTiO3 that are discussed in more detail in section 7. between the metastable configurations ðÞk ¼ 1...N À 1 and the ground state configuration (N) with Fk [ FN .This means rapid change rate of some FÀFk in some temperature 5 Thermodynamics of Critical Phenomena kBT ranges with respect to the limit of stability. Thermodynamics of critical phenomena is usually dis- Therefore, the thermodynamic criterion for instability cussed in terms of the instability of a homogeneous system and critical phenomena is that the entropies of derived from the combined law, i.e. Eq 2,asEq4 and metastable configurations are higher than that of the follows[11] stable configuration, but their differences are large enough that the stable configuration remains stable with respect to oYa ¼ 0 ðEq 23Þ the metastable configurations so that there are no first-order o a X transitions until the instability and critical point are The thermodynamic criterion of instability based on reached. entropy is written as oT ¼ 0 ðEq 24Þ oS 6 Examples: Anti-Invar Cerium, Invar Fe3Pt, and Al-Zn binary When the system approaches the limit of stability,[11] this derivative approaches zero, and its inverse, i.e., the Cerium (Ce) is a unique element. Its ground state config- change of the entropy of the system, diverges, i.e. uration under ambient pressure is non-magnetic face cen- oS tered cubic (fcc) and its stable configuration at room ¼þ1 ðEq 25Þ oT temperature is ferromagnetic fcc with some degree of The fluctuation of configurations reaches the whole magnetic disordering, while all other magnetic elements in system. After crossing the limit of stability, the system the chemical periodic table are opposite with their ground becomes inhomogeneous in the time and space scales state configuration being magnetic. Ce has a critical point under consideration. around TC = 480-600 K and PC = 1.45-2 GPa with the Differentiation of Eq 8 with respect to temperature gives upper bound commonly accepted in the literature with details discussed in Ref. 46.

123 J. Phase Equilib. Diffus.

In an effort to predict the thermodynamic properties of Ce, we first considered two configurations: non-magnetic (NM) and ferromagnetic (FM) with their Helmholtz ener- gies predicted by first-principles calculations using the partition function approach discussed above.[46] It should be noted that the entropy in the Helmholtz energy of each individual configuration contains the configurational con- tributions from lower scales, i.e. thermal electronic con- figurational contributions due to non-zero electronic density at the Fermi level and the vibrational frequency configurational contributions due to atomic vibrations as shown below[32] k F ¼ Ec þ Fvib þ Fel ðEq 28Þ where Ec is the static total energy at 0 K predicted by first- principles calculations, Fvib the lattice vibrational free energy, and Fel the thermal electronic contribution. The first term can be calculated directly using e.g. VASP [47] code. Fvib and Fel are related to contributions at finite temperatures with the same equation form as Eq 7. Fel, which is evaluated from the electronic densities of state at different volumes,[32] is important for metals due to the non-zero electronic density at the Fermi level. Fvib can be obtained from first-principles phonon calculations for accurate results[32,33] or the Debye model for the sake of simplicity using the modified scaling factor.[34] However, the consideration of two configurations was not able to reproduce the experimentally observed critical phenomena. This is because their fluctuations in the finite space result in the interactions of magnetic spins in dif- ferent orientations which are not captured by the two configurations considered. Following the conventional approach in the literature, an additional contribution to the free energy of the system in terms of ‘local-moment’ Fig. 2 (a) Total entropy, S, and vibrational entropy change, DSvib,of mechanism was added to account for the magnetic spin Ce at 0 GPa with T (K) and (b) T–V phase diagram of Ce, with details disordering, resulting in good agreement with experimental in Ref. 46 observations as shown in Fig. 2 for total entropy and T–V [46] phase diagram along with all other anomalies in the system ‘local-moment’ mechanism was corrected, and the T as detailed in Ref. 46 such as the critical point at TC ¼ critical point was predicted at C ¼ 546 K and P : 476 K and PC ¼ 2:22 GPa. C ¼ 2 05 GPa, closer to the commonly accepted values The next step is to add an antiferromagnetic (AFM) than our previous predictions. The key is that the interac- configuration. The first-principles calculations predict that tions between two configurations when they fluctuate is the static energy of the AFM configuration is between included in the thrid configuration. those of NM and FM configurations.[48] Since the inter- The underlying of our ‘itinerant-electron’ [48] action due to magnetic spin disordering is included in the magnetism model closely resembles the Kondo-Ander- AFM configuration, the ‘local-moment’ mechanism was son magnetic model as summarized by Kouwenhoven and [49] no-longer needed. The fluctuation of the NM, AFM, and Glazman. At low temperatures, the Kondo-Anderson FM configurations enabled the predictions of all anomalies model states that impurity magnetic moment and one in the system as shown in Fig. 3 for various entropy conduction electron moment bind and form an overall changes and T-P phase diagram in addition to Schottky nonmagnetic configuration, while at high temperatures, the anomaly for heat capacity. Particularly the overestimated binding is broken resulting in a magnetic entropy of entropy change of the transition along the phase boundary approximately kB ln 2. In our itinerant-magnetism model in our previous work with two configurations only and the for the Ce transition, the nonmagnetic is the occupied

123 J. Phase Equilib. Diffus.

Fig. 4 Thermal populations, i.e. pk, of the nonmagnetic (red dot- dashed), anti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a function of temperature at the critical pressure of 2.05 GPa from Ref. 48

thermodynamically guaranteed. It depends on the volumes of the metastable configurations.[50,51] For example, the isobaric volume curves in the T–V oV phase diagram of Invar Fe3Pt in Fig. 5(b) show oT )À1 oV at the critical point marked by the green circle and oT \0in a considerable temperature ranges away from the critical point. In this system, a supercell with 9 Fe and 3 Pt atoms was considered, resulting in 29=512 magnetic configura- tions with 37 of them being unique due to the symmetry. This negative thermal expansion or thermal contraction is because the metastable magnetic spin-flip configurations Fig. 3 (a) Predicted entropy changes in terms of lattice vibration have smaller molar volumes than the FM configuration in (lat), lattice vibration plus thermal electron (lat ? el), and lattice vibration plus thermal electron and plus configuration coupling Fe3Pt. When the probability of the metastable magnetic (lat ? el ? f) changes along the phase boundary, (b) Predicted T–P spin-flip configurations increases with temperature due to phase diagram, both in comparison with experimental data with their higher entropies than that of the FM configuration, the details in Ref. 48 oV thermal expansion of the system, i.e. oT, is the result of the competition between the positive thermal expansion of ground configuration and the magnetic configurations are each configuration and the volume reduction due to the empty at 0 K. At finite temperatures, our model has a replacement of the FM configuration by the magnetic spin- magnetic entropy term as shown by Eq 7, slightly different flip configurations. When the latter is larger than the for- from the simple Kondo-Anderson model. At high enough mer, the thermal expansion of the system becomes nega- temperatures the mixture between the FM and AFM con- tive. These anomalies can be generalized to a wide range of figurations results in a magnetic entropy of  k ln 2 as B emergent behaviors for all physical properties of a system, shown by the populations of configurations in Fig. 4. when a system exhibits properties that its constituents do It should be noted that the changes of all physical not possess such as negative thermal expansion in Fe Pt properties diverge at the critical point as shown by Eq 25. 3 discussed above, with respect to composition and elastic, For example, Fig. 5(a) plots the predicted T–V phase dia- electric, magnetic fields, and their combinations with the gram of Ce with isobaric volume curves and available [14,44,45,51] oV present model as the theoretical foundation. experimental data superimposed. It can be seen that oT ) In both Ce and Fe3Pt cases, the configurations in the þ1 at the critical point marked by the green circle, and k 2 fg1; ...; m scale are various magnetic configurations significant anomaly exists at pressures considerably away with two sets of sub-configurations, i.e. thermal electronic from the pressure at the critical point. Since V and T are not and vibrational configurations, in the l1 2 fg1; ...; n1 and conjugate variables in the combined law of thermody- l2 2 fg1; ...; n2 scales. Based on the discussions in sec- [11] oV namics (see Eq 2), the positive sign of oT is not tion 3, it is possible to construct two new sets of thermal electronic and vibrational configurations averaged across

123 J. Phase Equilib. Diffus.

phenomena needs to be further improved with multiple configurations as discussed above.[45]

7 Unresolved Issues in ABO3 Perovskites

ABO3 perovskites represent a group of ferroelectric materials that directly convert electrical energy to mechanical energy or vice versa. Many of them experience ferroelectric-paraelectric (FE-PE) phase transitions as a function of temperature or electric field. As an insulator,

lead titanate (PbTiO3, PTO) is a benchmark ferroelectric material. The between its two known perovskite structures, i.e. the ferroelectric tetragonal phase and the para-electric cubic phase,[53] occurs at about 763 K at ambient pressure and at room temperature at about 12 GPa[54,55] and has been considered as a classical example to understand structural and ferroelectric phase transitions in perovskites.[44,54,56–58]

The ferroelectricity in the tetragonal PbTiO3 phase is due to the displacement of Ti atoms away from the center position in the tetragonal structure, resulted in spontaneous electric polarization. While the FE-PE phase transition is regarded as the typical displacive transition associated with softening of the relevant phonon modes, a certain degree of order–disorder mechanism has been considered in the literature associated with the orientation change of local distortions.[59,60] The direct calculation of the free energy of the para-electric cubic PbTiO phase in terms of Eq 28 is not possible because its Fig. 5 Predicted temperature–volume phase diagrams with the 3 miscibility gap (yellow shaded area) and critical point (green circle) vibrational free energy cannot be evaluated due to its insta- [8] marked, (a) cerium and (b) Fe3P. Additional isobaric volumes are bility at zero K and ambient pressure. On the other hand, x-ray plotted for several pressures with available experimental data absorption fine-structure (XAFS) measurements by Sicron superimposed et al.[59] showed that the Ti atoms are displaced relative to the oxygen octahedra cage center both below and above the FE- various magnetic configurations so that the thermal elec- PE phase transition temperature. tronic and vibrational properties of the system can be Through AIMD simulations shown in Fig. 7, Fang predicted. The corresponding results are being investigated [44] et al. observed that the amount of tetragonal configu- and will be published separately. ration decreases with the increase of temperature and Let us consider the Al-Zn binary system with its phase reaches a plateau slightly below 50% above the FE-PE diagram under ambient pressure shown in Fig. 6(a) calcu- transition temperature in the cubic PbTiO phase region lated from the CALPHAD database in the literature[52] with 3 with this transition temperature defined by the x-ray a miscibility gap in the fcc phase. When the Gibbs energy diffractions with lower time and spatial resolutions than is plotted with respect to the composition of Zn in [54] those of XAFS (Fig. 7a). Figure 7(b) also shows that Fig. 6(b), its curvature, i.e. the chemical potential deriva- about 80% of the tetragonal configurations in the cubic tive with respect to composition, changes sign in accor- phase region are polarized though the over-all polarization dance with the stability criterion that the derivative of a [44] is almost zero. The other half of the Ti atoms in the potential with respect to its conjugate molar quantity [11] cubic configuration are on transition from one to another changes sign at the limit of stability. However, the plot [45] polarization directions at any moment of time. This of the with respect to entropy in observation confirms our theory as discussed in the previ- Fig. 6(c) does not change its curvature as it should when ous section for Invar Fe Pt and anti-Invar Ce, i.e. high- crossing the stability boundary based on Eq 24.This 3 temperature structures are dynamic averages of low-tem- indicates that the present CALPHAD modeling of critical perature configurations.

123 J. Phase Equilib. Diffus.

b Fig. 6 (a) Calculated temperature-composition phase diagram of Al- Zn under ambient pressure from the CALPHAD database by Mey [52]; (b) Gibbs energy of the fcc phase as a function of Zn content at 373 K o2G shown that the curvature of the curve, i.e. 2, changes sign with oðÞxZn respect to Zn content with the region of negative curvature being unstable; (c) Internal energy of the fcc phase as a function of entropy o2 at mole fraction of Zn equal to 0.35 shown that U does not change oS2 sign with respect to entropy. The numbers on the curve are used to calculate the temperature in Kelvin using the function shown above the figure

frequencies)[61] using the direct (or supercell, small dis- placement, frozen phonon) method, we developed an approach that combines the accurate force constants cal- culated by the direct method in real space and the long- range dipole–dipole interactions calculated by linear response theory in reciprocal space.[62,63] To address the imaginary phonon modes predicted by first-principles cal-

culations for ideal cubic ABO3 perovskites, we used a pseudo-cubic supercell, i.e. with the shape of the supercell being cubic, but allowing the internal atomic positions fully relaxed, to successfully predict the experimentally [64] observed phonon dispersion curves of SrTiO3 and [65] PbTiO3, but this approach can only predict first-order transition and cannot predict the second-order transition and the critical point. Therefore, the challenge remains how to define the low-

temperature configurations for ABO3 in general and PbTiO3 in particular. For Ce and Fe3Pt, consideration of magnetic configurations with spins in two directions was

sufficient. For PbTiO3, it seems that the polarizations in six directions are needed based on the results from AIMD simulations,[44,45] resulted in a huge number of configura- tions even for a 40 atom supercell with 8 Ti atoms, i.e. 68 ¼ 1; 679; 616 configurations. It is self-evident that a new strategy is needed to discover those low energy, rel- evant configurations, and we are actively working on this topic.

8 Information Entropy and Internal Processes

Szilard[66,67] was probably the first to make a link between entropy and information when he tackled the famous Maxwell’s demon through thought experiments. He rigor- ously demonstrated that a ‘‘biological phenomenon’’ of a nonliving device generates the same quantity of entropy required by thermodynamics, a critical link in the inte- gration of physical and cognitive concepts that later In order to be able to predict the phonon properties of became the foundation of and a key insulating polar materials with LO-TO splitting (splitting idea in complex systems. In investigating the mathematical between longitudinal and transverse optical phonon solution to communication and message transmission

123 J. Phase Equilib. Diffus.

and proposed that information corresponds to a negative term in the final entropy of a physical system as follows[70]

Sfinal ¼ Sinitial À I ðEq 29Þ

where Sinitial and Sfinal are the initial and final entropies of the system, and I the information of the system. Brillouin’s idea of dealing with information and physical entropy on an equal basis has been widely accepted in the commu- nity.[74] The exchange between energy and information was further elaborated by Landauer[75,76] who perceived that a logically irreversible process reduces the degrees of freedom of the system, thus entropy, and therefore must dissipate energy into the environment, hence erasing information in memory entails entropy increase in the environment. This is commonly referred to as Landauer’s erasure principle, also investigated theoretically by Ben- nett[77] and more recently verified experimentally.[78,79] Let us use the IP discussed in the present paper to understand the concept of information and its generationÀÁ or n loss. This IP may consume some masses/nutrients dNi , generate some output masses/wastes dNw and heat ÀÁ j dipQ , and re-organize itsÀÁ configurations to produce cer- tain amount of information dipI , resulting in the entropy production with a similar form as Eq 1 d Q X X d S ¼ ip À SndNn þ SwdNw À d I ðEq 30Þ ip T i i j j ip n w where Si and Sj are the entropies of nutrient i and waste j, respectively. Let us study several thought experiments of spontaneous IP, i.e. d S [ 0: Fig. 7 (a) Temperature dependence of the lattice parameters a, c of ip [44] PbTiO3 unit cell, crossed symbols from AIMD simulations, the a. dNn ¼ dNw ¼ 0, the IP behaves like a closed system open symbols from the XAFS measurements[59,86] and the closed i j symbols are from the x-ray diffraction,[54] (b) Fractions of the (i.e., involving no nutrient consumption and no waste tetragonal (closed squares) with and without polarization and cubic generation) (closed circles) and configurations as a function of temperature from [44] the AIMD simulations a:1. dipI [ 0, the IP produces more information, and the sub-system must have problems, Shannon[4,68] proposed the concept of informa- dipQ [ TdipI [ 0, i.e. the IP must generate tion entropy as the measure of the amount of information heat to be dissipated out to its surroundings, in that is missing before reception or before communication is accordance with the Landauer’s erasure prin- achieved and defined the information entropy as a macro- ciple. The more information is reflected by the state (a source) with the number of possible microstates more distinct configurations, such as formation of crystal from its liquid and data (ensembles of possible messages) that could be sent by that [68,80–82] source. Thus, information in the communication begins writing; with a set of messages, each with a probability, and the a:2. dipI\0, with information loss and average information content per message is defined in dipQ [ TdipI. Since TdipI is negative, dipQ analogy with Eq 7. could be either positive or negative. Brillouin extensively studied the relations between a:2:1. dipQ\0: the IP absorbs heat from the thermodynamics and information and suggested that outside of the sub-system, and the less information can be quantified using a system’s information is reflected by the resulted [69–73] entropy. He noted that entropy itself measures the fewer distinct configurations, such as lack of information about the actual structure of a system

123 J. Phase Equilib. Diffus.

melting of crystal into its liquid and the system, the information change of the system is fully dic- data erasing; tated by the internal processes in the system which are a:2:2. dipQ [ 0: the IP produces more heat, regulated by the heat and mass exchanges between the resulting in self-destruction, such as fire. system and its surroundings that control the available nutrients for internal processes, as shown by Eq 1 and 30. b. dNn [ 0 and dNw [ 0, the IP behaves like an open i j The total information change of a system would be the system (i.e., involving nutrient consumption and waste sum of individual internal processes. Following the dis- n w generation). By definition, neither dNi nor dNj can be cussions by Shannon[4,68] and Brillouin[69–73] we can re- negative as their signs are pre-determined by the signs write Eq 8 and 9 as follows in front of them in Eq 30. Xm P P k k k n n w w nw S ¼ÀI þ p S ðEq 32Þ b:1. À Si dNi þ Sj dNj ¼ dS [ 0, more waste k¼1 entropy than nutrient entropy, and d Q [ Td I À ip ip Xn nw TdS for a spontaneous IP Sk ¼ÀIljk þ pljkSljk ðEq 33Þ l¼1 b:1:1. dipI [ 0: the IP either must produce heat if Xm Td I [ TdSnw as in a.1 or produce/absorb ip Ik ¼ÀSconf ¼ k pk ln pk ðEq 34Þ heat which importantly indicates that the IP B k¼1 may produce more information through Xn intensive metabolism without generating ljk ljk ljk I ¼ kB p ln p ðEq 35Þ heat or with reduced heat generation; l¼1 b:1:2. d I\0: the IP can either absorb or produce ip where Ik and Iljk denote the information in the system level, heat as in the case in a.2. P P i.e., scale k 2 fg1; ...; m , and the sub-scale level of n n w w nw b:2. À Si dNi þ Sj dNj ¼ dS \0, lower waste l 2 fg1; ...; n , respectively. The sub-scale information entropy than nutrient entropy, and dipQ [ TdipI À affects the system level information through its contribu- TdSnw for a spontaneous IP. tion to the probability pk as shown by Fk in Eq 21. As all spontaneous internal processes produce entropy, b:2:1. d I [ 0: the IP must produce heat as in a.1, ip one may tend to think that the information of the universe but with the amount being Td I TdSnw, ip À has been decreasing from the beginning of time if the nw larger than TdipI by ÀTdS , such as the beginning of time could be defined such as by the Big Bang growth of young organisms; though certain sub-systems may experience an increase of b:2:2. dipI\0: the IP must produce heat if their information as discussed in the thought experiments. nw TdipI [ TdS and may produce/absorb heat As the sub-systems are brought across their limits of sta- nw if TdipI\TdS as in the case in a.2, noting bility through the interactions with their surroundings, self- nw that dS \0. organized structures result. This is what discussed by P P [15] n n w w nw Kondepudi and Prigogine, that instability in a system b:3. À Si dNi þ Sj dNj ¼ dS ¼ 0, a steady enables the generation of dissipated structures, thus more state in terms of balanced entropy between nutri- distinct configurations, as also demonstrated in three ents and wastes. This case is similar to those in a.1 examples in section 6. Therefore, the fundamentals of and in a.2, but with a constant of mass flow in and thermodynamics and information discussed in the present out of the spontaneous IP. paper provide a framwwork for investigations of complex From Eq 30, one can further write the information systems such as nano devices,[83] quantum comput- [82,84] [6,85] change for an irreversible process (dipS [ 0) as follows ing, and ecosystems. d Q X X d I\ ip À SndNn þ SwdNw ðEq 31Þ ip T i i j j 9 Summary which gives the upbound of information that can be pro- duced by an internal process. It can be seen that the The fundamentals of entropy are discussed in this paper information generation (loss) can be increased (decreases) emphasizing its statistic nature, configurations, and infor- with higher heat production, lower entropy of nutrient mation in comparable time and space scales. It is pointed inputs, and higher entropy of waste outputs. It should be out that the entropies of individual configurations play an pointed out that even though the entropy change due to essential role in determining their statistic probabilities and internal processes is only part of total entropy change of the thus the configurational entropy and information of the

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